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In this paper, we consider the initial boundary value problem of the twodimensional multi-term time fractional mixed diffusion and diffusion-waveequations. An alternating direction implicit (ADI) spectral method is developedbased on Legendre spectral approximation in space and finite differencediscretization in time. Numerical stability and convergence of the schemes areproved, the optimal error is $O(N^{-r}+\tau^2)$, where $N, \tau, r$ are thepolynomial degree, time step size and the regularity of the exact solution,respectively. We also consider the non-smooth solution case by adding somecorrection terms. Numerical experiments are presented to confirm ourtheoretical analysis. These techniques can be used to model diffusion andtransport of viscoelastic non-Newtonian fluids.

An alternating direction implicit spectral method for solving two  dimensional multi-term time fractional mixed diffusion and diffusion-wave  equations